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Chapter 9. Supply Chain Models

This is an introduction chapter quotation. It is offset three

inches to the right.

9.1. Introduction

Model Characteristics Multicommodity

Multi-echelon

Capacitated Facilities

Capacitated Channels

Deterministic or Stochastic

Single and Multiple Periods

Model and Algorithm Hierarchy

Evaluate or Benchmark

Digital simulation or spreadsheet or maps

Distribution Channel Selection

Current facilities

Select best distribution channel and set inventory levels

1 ● Chapter 9. Supply Chain Models Logistics Systems Design

Spreadsheet or custom programming

Network Optimization

Current facilities and capacities

Optimize the product flow

Linear Programming solver

Location-Allocation

Moving Facilities

Current facility status

No site-dependent costs

Cost proportional to material flow quantity, such as transportation and material handling in the

distribution center

Alternative generating algorithm

Limited location resolution (at the level of a county or metropolitan area)

Approximate algorithms

Non-linear optimization, specialized heuristics

Production-Distribution

Everything is allowed to change

Alternative selecting

Site dependent costs

Mixed Integer Programming solver

Production-Distribution Models and Algorithms

Kuehn and Hamburger

Drop - Add- Swap heuristic

Logistics Systems Design Chapter 9. Supply Chain Models ● 2

Geoffrion and Graves

Benders’ decomposition

Goetschalckx and Wei

Disaggregated MIP Formulation with Branch and Bound with LP relaxation

9.2. Transportation Mode Selection Model

Parameters and Variables

HCF inventory holding cost factor (dollars per dollar of inventory per year)

pD annual demand for product p

pv value of a unit of product p

ppc unit production cost of product p

mTT transit time of transportation mode m (expressed in years)

mptc transportation cost for shipping one unit of product p with transportation mode m

mpTB transportation batch size of product p shipped with transportation mode m

psc annualized fixed storage cost per unit of product p

pPC total annual production cost for product p

mpTC total annual transportation cost for product p shipped with transportation mode m

mppic unit pipeline inventory cost of product p shipped with transportation mode m

mpPIC pipeline inventory cost of product p shipped with transportation mode m

impOCIC cycle inventory cost of product p shipped with transportation mode m from plant i

jmpDCIC cycle inventory cost of product p shipped with transportation mode m to distribution center j

pd average demand during observable demand period, e.g. daily demand, of product p

pVd variance of the demand during observable demand period, e.g. daily demand, of product p

pCVd coefficient of variation of the demand during observable demand period, e.g. daily demand, of product p


mpLT average lead time for the delivery of product p using transportation mode m expressed in observable demand periods, e.g. days

mpVLT variance of the lead time for the delivery of product p using transportation mode m expressed in observable demand periods, e.g. days

mpSI safety inventory of product p shipped with transportation mode m

jmpSSIC safety stock inventory cost in distribution center j of product p shipped with transportation mode m

jmpFSC total annual fixed storage cost in distribution center j for product p shipped with transportation mode m

pTIC total invariant cost for product p

mpTVC total variable cost for product p using transportation mode m

mpTFC total fixed cost for product p using transportation mode m

mpTAC total aggregate cost for product p using transportation mode m

A typical value of the holding cost factor or HCF is 0.25, i.e., 25 cents per dollar of inventory per year.

To find the holding cost factor for another time period, we have to divide by the number of time periods

in a year. A HCF equal to 0.25 per year equals a holding cost factor of 0.25/365 = 0.000685 per day.

Demand During Lead Time

Prob

abilit

y

Demand

d.LT

CD=95 %

kσ

Figure 9.1. Demand During Lead Time Distribution

Assuming single sourcing and a single transportation mode of the supply of each product at the

distribution centers, the safety inventory at the distribution center can then be computed as


SI k LT Var d Vard= ⋅ ⋅ + ⋅2LT (9.1)

CVVard

Var CV d

dd

d d

=

= ⋅b g2 (9.2)

SI k LT CV Var dd LT= ⋅ ⋅ + ⋅2 (9.3)

This formula quantifies the common practice in industry of keeping a safety stock level on hand equal to

a number of demand periods. The formula shows the relationship between the safety inventory and the

customer service level based on probability of delivery out on-hand inventory, the average and variance

of the lead time, and the average and variance of the demand. Since the demand is determined by the

customers and the service level is usually a mandate from corporate management, the only factors that

can be influenced by the warehouse manager are the average and variance of the lead time and the

variance of the demand. Safety inventory can be reduced if the input and output flows, i.e., supply and

demand, are kept as constant as possible and if the lead time for replenishments is reduced.

Cost Computations

Invariant Costs

Invariant costs are costs that are incurred by the logistics systems but that do not depend on the selection

of the distribution channel. Since, it is assumed that the total demand does not depend on the

distribution channel, the total production cost is a member of the invariant costs. The total production

costs is computed as product of the yearly demand and the unit production costs.

PC D pcp p= ⋅ p (9.4)

TIC PCp p= (9.5)

Fixed Costs

Fixed costs are costs whose magnitude does not change during the operation of the distribution channel.

The fixed costs may be different from one distribution channel to another, but once a channel selection

has been made, the fixed costs remain unchanged. In distribution channel design, the size of the

warehouse remains unchanged for extended periods of time. Hence, the annualized cost per warehouse

location or per cubic foot of warehouse space is a component of the fixed costs. The space occupied by


a product is typically computed as proportional to the maximum inventory of this product. The

maximum inventory for each product is the sum of the cycle and safety inventory. The total fixed

storage cost is then the product of the maximum product inventory and the annualized unit storage cost

for that product.

jmp mp mp pFSC TB SI sc = + ⋅ (9.6)

mp jmpTFC FSC= (9.7)

Variable Costs

The variable costs are costs that may change during the operation of the distribution channel. Typically

these costs are a function of either the transfer batch size or order quantity and of the total annual

demand.

The total transportation cost is computed as the product of the annual demand and the unit transportation

cost.

mp p mpTC D tc= ⋅ (9.8)

The pipeline inventory cost is computed as the product of the annual demand, the value of single unit at

the origin of the pipeline, the transit time for the flow to go through the pipeline, and the inventory

holding cost rate.

mp p mpic v TT HCF= ⋅ ⋅ (9.9)

mp p p mPIC D v TT HCF= ⋅ ⋅ ⋅ (9.10)

When shipping in transportation batches, there is both an inventory build up at the source until the

shipment occurs, and inventory depletion at the destination starting after a replenishment shipment

arrives. If we assume a constant build up and depletion rate, then the average inventory follows the

classical saw tooth pattern and the average inventory is half the maximum inventory or transportation

batch size. The value of one unit of product is different at the origin and destination. At the destination

the corporation has invested the sum of the production cost, the transportation cost, and the in-transit

inventory cost in the product. The origin and destination cycle inventory costs are then computed as the

product of half the transportation batch size, the holding cost rate, and the appropriate unit value.


( )2imp mp pOCIC TB v HCF= ⋅ ⋅ (9.11)

( ) ( )2imp mp p mp mpDCIC TB v tc pic HCF= ⋅ + + ⋅ (9.12)

The safety inventory level is computed with the assumption of a linear safety inventory policy as

expressed in (9.3). The safety inventory cost is then computed as the product of the safety inventory

level, the value for each unit, and the holding cost rate.

2mp mp p mp pSI k LT CVd VLT d= + (9.13)

(mp mp p mp mpSIC SI v tc pic HCF= ⋅ + + ⋅)

mp

(9.14)

The total variable cost is computed as the sum of the transportation cost, the pipeline inventory cost, the

origin and destination cycle inventory costs, and the safety inventory cost.

mp mp mp imp jmp jmpTVC TC PIC OCIC DCIC SIC= + + + + (9.15)

Finally, the total aggregate cost is computed as the sum of the total invariant, variable, and fixed costs.

mp p mpTAC TIC TVC TFC= + + (9.16)

Ballou Channel Selection Example

The calculations for the channel selection example in Ballou (1998) are summarized in the next table.

The safety inventory levels in the plant and the distribution center are not computed with the standard

formulas shown above, put are based on description given in Ballou. For example, the average

inventory in the distribution center when shipping by piggyback is specified as 0.93 * 50,000 = 46,500.

Since the average cycle inventory is 35,000 / 2 = 17,5000, the average safety inventory is then 46,500 –

17,500 = 29,000. The safety inventory levels for the other transportation modes were calculated in a

similar way.

There were no warehouse storage costs included in the example. Based on the given parameters, truck

transportation is the preferred distribution transportation mode.


Table 9.1. Channel Selection Calculations for the Ballou Example

Annual Demand 700000Annual Holding Cost Rate 0.3Unit Production Cost $30.00Unit Annualized Warehouse Cost $0.00

Rail Piggyback Truck AirUnit Transportation Cost ($) 0.1 0.15 0.2 1.4Channel Transit Time (days) 21 14 5 2Transportation Batch Size 70000 35000 35000 17500

Rail Piggyback Truck AirProduction Cost $21,000,000 $21,000,000 $21,000,000 $21,000,000Total Invariant Costs $21,000,000 $21,000,000 $21,000,000 $21,000,000Transportation Costs $70,000 $105,000 $140,000 $980,000In-Transit Inventory $362,466 $241,644 $86,301 $34,521Order Fequency per Year 10 20 20 40Order Cycle Length in Years 0.1 0.05 0.05 0.025Plant Max Cycle Inventory 70,000 35,000 35,000 17,500Plant Cycle Inventory Costs $315,000 $157,500 $157,500 $78,750Plant Safety Inventory 65,000 29,000 24,500 11,250Plant Safety Inventory Costs $585,000 $261,000 $220,500 $101,250Unit Value at DC $30.62 $30.50 $30.32 $31.45DC Max Cycle Inventory 70,000 35,000 35,000 17,500DC Cycle Inventory Costs $321,487 $160,100 $159,197 $82,554DC Safety Inventory 65,000 29,000 24,500 11,250DC Safety Inventory Costs $597,047 $265,308 $222,876 $106,141DC Maximum Inventory 135,000 64,000 59,500 28,750Total Marginal Costs $2,251,000 $1,190,552 $986,375 $1,383,216Annualized Warehouse Costs $0 $0 $0 $0Total Variant Costs $2,251,000 $1,190,552 $986,375 $1,383,216Total Cost $23,251,000 $22,190,552 $21,986,375 $22,383,216

9.3. Kuehn and Hamburger Model


Zero echelon

Uncapacitated depots


Depot Handling Cost possible as an extension

Deterministic

Single Period

Arc or path formulation are equivalent since this is a zero echelon model.

“Weak” formulation with aggregate consistency or linkage constraints.

Model

min

. .

{ , },

z f y c x

s t x i

x My

y x

j j ij iji

M

j

N

ijj

N

ij ji

M

j ij

= +

j

FHG

IKJ

=

− ≤

∈ ≥

==

=

=

∑∑

∑

∑

11

1

1

1

0

0 1 0

∀

∀

}i

(9.17)

Shadow Prices and Site Relative Costs

iu Current cost for servicing customer i

Error! Objects cannot be created from editing field codes. Site relative cost for opening warehouse j based on the current customer service cost . Note that both and are the cost for servicing the total demand of a customer for a particular product.

iu iu ijc

{1

( ) min 0,M

j j iji

f c uρ=

= + −∑U (9.18)

Drop-Add-Swap Heuristic

Drop

Start with all facilities open

Close the one with the largest positive relative site cost

Until no further decrease in costs


Add

Start with all facilities closed

Open the one with the most negative relative site cost

Until no further decrease in costs

Swap

Start with drop

Swap (Open and Close) the two facilities with the most extreme relative site cost

Until not further cost reduction

Drop-Add-Swap Heuristic Example

The drop-add-swap heuristic is applied to the supply chain configuration example in Ballou (1998),

pp. 498. This heuristic only can handle zero echelon supply chains, where goods flow directly from the

source to the sink facilities, so the manufacturing plants are ignored in the following computations. This

heuristic can only handle uncapacitated source facilities, so the capacity of warehouse W1 is ignored.

The fixed costs are $100,000 and $500,000 for warehouse 1 and 2, respectively. The cost to serve a

customer is the sum of the transportation cost between the respective origin and destination facility and

the handling cost in the warehouse. Let u and be the current unit cost to serve customer i and the

marginal unit cost to serve customer i from warehouse j, respectively. Let U and be the current total

cost to serve customer i and the total marginal cost to serve customer i from warehouse j, respectively.

Note that this is a change in notation from the original notation used to define the site relative cost

factor. The marginal costs can then be summarized in the Table 2.

i ijc

i ijC

The drop phase starts with the current customer service costs based on the actual situation or the best

current cost for serving each customer, based on the solution of a standard network flow problem for

each of the products with as sources all warehouses and as sinks all customer-product combinations.

However, since the source facilities are uncapacitated, we can easily find the solution to the network

flow problem since each customer will always be served from the cheapest open source.


Table 9.2. Drop Phase Marginal Costs for Servicing Customers

Destination C1P1 C2P1 C3P1 C1P2 C2P2 C2P2Demand 50,000.00 100,000.00 50,000.00 20,000.00 30,000.00 60,000.00

OriginW 1 c i1 4+2=6 3+2=5 5+2=7 3+2=5 2+2=4 4+2=6

C i1 300,000.00 500,000.00 350,000.00 100,000.00 120,000.00 260,000.00W 2 c i2 2+1=3 1+1=2 2+1=3 3+1=4 2+1=3 3+1=4

C i2 150,000.00 200,000.00 150,000.00 80,000.00 90,000.00 240,000.00U i 150,000.00 200,000.00 150,000.00 80,000.00 90,000.00 240,000.00

Since at the beginning of the drop phase both distribution centers are assumed to be open, the best

current cost for servicing a customer is the minimum of the service cost from either warehouse. For this

particular example at this particular iteration, the least expensive service is to deliver to all customers

from warehouse W2. The total cost for the configuration at this time is then the sum of the fixed costs

for all warehouse plus the sum of the current best service costs, or

( ) ( )100000 500000 150000 200000 150000 80000 90000 240000600000 910000 1510000

TC = + + + + + + +

= + =

Note that the fixed cost of warehouse W1 is still included in the total cost even though it is not used.

The next step is to compute the site relative cost factors for each warehouse site based on the current

values of the customer service costs. With the notation adopted in this example, the site relative cost

factors are computed with the following formula

( ) { }1

min 0,M

j j iji

f Cρ=

= + −∑U iU

{ } { }{ } {{ } {

1

min 0,300000 150000 min 0,500000 200000

100000 min 0,350000 150000 min 0,100000 80000

min 0,120000 90000 min 0,360000 240000

100000 0 100000

ρ

− + −

= + − + − + − + −

= + =

}}

+

{ } { }{ } {{ } {

2

min 0,150000 150000 min 0,200000 200000

500000 min 0,150000 150000 min 0,80000 80000

min 0,90000 90000 min 0,240000 240000

500000 0 500000

ρ

− + −

= + − + − + − + −

= + =

}}

+

The drop heuristic will close the facility with the largest positive site relative cost, which in this case is

warehouse W2 with a site relative cost of 500,000. The total cost for this new configuration is

$1,830,000, as computed by the next formula.


( ) ( )100000 300000 500000 350000 100000 120000 360000100000 1730000 1830000

TC = + + + + + +

= + =

This is more expensive than the previous configuration of the drop phase and so the drop phase would

not close warehouse W2 and the drop phase terminates.

If closing warehouse W2 would have reduced the total cost, then the customer service costs have to be

recomputed based on the current configuration. Since the only warehouse facility open at this time

would be W1, the updated customer service costs would be the marginal service costs from facility W1.

At this point only the single warehouse W1 would remain open and the drop heuristic would terminate.

In larger supply chain systems the drop phase may go through several iterations, closing during each

iteration the facility with the largest positive site relative cost factor and then recomputing the customer

service costs.

The add phase starts with the current customer service costs based on the actual situation or with a single

open distribution center. From earlier calculations, the best warehouse to open based on the marginal

customer service costs is depot W2. The customer service costs are shown in Table 2. The site relative

costs based on these customer service costs have already been computed and none of the site relative

costs are negative, so the add heuristic does not add a facility and terminates. The total cost at the end of

the add phase is $1,410,000.

( ) ( )500000 150000 200000 150000 80000 90000 240000500000 910000 1410000

TC = + + + + + +

= + =

The swap phase attempts to improve the total cost by executing first the drop phase and then closing one

facility and simultaneously opening one additional facility. The facility selected for closure is the open

facility with the largest positive relative site cost factor. The facility selected for opening is the closed

facility with the smallest relative site cost factor. For this particular example, this would entail closing

warehouse W2 and not opening an additional warehouse since all warehouses are still open. This

configuration has already been examined and the total cost would be $1,830,000 and the swap phase

terminates without making a single exchange.

The overall heuristic terminates with the best configuration after the swap phase, which in this particular

example consists of warehouse W2 open and servicing all customers with all products and warehouse

W1 is closed for a total cost of $1,410,000.


If the add phase would have been started with the single warehouse W1 open, then the total cost would

have been $1,830,000 and the site relative cost for warehouse W2 would have been (negative) -320,000

and warehouse W2 would have been added. The new configuration would have a total cost of

$1,510,000 and would have all warehouses open, which is equivalent tot the starting configuration of the

drop phase. The swap phase would not be able to make any exchanges and the heuristic terminates with

both warehouses open, which is clearly not optimal. Obviously, this example is too small to

demonstrate the normal iterations of the Drop-Add-Swap heuristic.

9.4. Arc-Based Model Solved with Mixed Integer Programming

M1

M2

M3

C1

C2

C3

S1

S2

S3

W1

W2

W3

W4

Figure 9.2. Arc-Based Multi-echelon Supply Chain Example Illustration


Figure 9.3. Arc-Based Supply Chain Example Transportation Data

Figure 9.4. Arc-Based Supply Chain Example Facility Capacity and Cost Data


Figure 9.5. Arc-Based Supply Chain Example Transportation Supplier-Manufacturing Model

Figure 9.6. Arc-Based Supply Chain Example Transportation Manufacturing-Customer Model


Figure 9.7. Arc-Based Supply Chain Example Facility Capacity Model

Figure 9.8. Arc-Based Supply Chain Example Solver


Figure 9.9. Arc-Based Supply Chain Example Transportation Supplier-Manufacturing Solution

Figure 9.10. Arc-Based Supply Chain Example Transportation Manufacturing-Customer Solution


Figure 9.11. Arc-Based Supply Chain Example Facility Capacity Solution

9.5. Geoffrion and Graves Distribution Model


Single echelon

Capacitated Depots (Lower and Upper Bound)

Depot Handling Cost

Deterministic

Single Period

Path Formulation


Depot Single Sourcing

“Weak” Formulation

Additional Linear Constraints in z and y

Model

min

. .

c x f z h r y

s t x s ip

x r y

y k

TL z r y TU z j

ijkp ijkp j j j kp jkkpjijkp

ijkpjk

ip

ijkpi

kp jk

jkj

j j kp jkpk

j j

+ +

jk

FHGG

IKJJ

≤ ∀

=

= ∀

≤ ≤

∑∑∑

∑

∑

∑

∑

1

∀

∀

(9.19)

Benders (Primal) Decomposition Fix binary variables z and y

Solve independent commodity network flow problems

Determine total transportation cost

Add total transportation cost as a cut to binary master problem

Solve binary master problem

Iterate


Primal BinaryMaster Problem

PrimalNetwork FlowSubproblems

Transportation CostAdditional Constraint

Binary Variablesz and y

Figure 12. Benders Decomposition Flowchart

Primal Network Flow Subproblem

min

. . [ ]

[ ]

c x

s t x s v

x r y u

x

ijkp ijkpijkp

ijkpjk

ip ip

ijkpi

kp jk jkp

ijkp

∑

∑

∑

≤

=

≥ 0

(9.20)

Note that the y are parameters in this formulation, not variables.

Dual Network Flow Subproblem

max

. .

,

y v s u r y

s t u v c ijkp

v u unrestricted

ip ip jkp kp jkjkpip

jkp ip ijkp

ip jkp

0

0

= − +

− ≤ ∀

≥

∑∑

(9.21)

Note that the y are parameters in this formulation, not variables.

This formulation can also be written in function of the extreme points of the constraint polyhedral.

max,v u E

ipe

ip jkpe

kp jkjkpipip

ejkpe

y v s u r∈

= − + ∑∑0 y (9.22)

This yields the following constraints generated by the network flow subproblem for the primal integer

master problem.


y v s u ripe

ip jkpe

kp jkjkpip

0 ≥ − + ∑∑ y

E

(9.23)

Primal Binary Master Problem

min

. .

..

, { , }

f z h r y y

s t y k

TL z r y TU z j

y v s u r y e

z y

j j j kp jkkpj

jkj

j j kp jkpk

j j

ipe

ip jkpe

kp jkjkpip

+FHGG

IKJJ

+

= ∀

≤ ≤ ∀

≥ − + =

∈

∑∑

∑

∑

∑∑

0

0

1

1

0 1

(9.24)

Example Consider the problem of designing the strategic production and distribution network. There are two

products, two manufacturing plants and three customers. There are also two potential distribution center

sites. The potential network is shown in Figure 13. A customer can only be served by a single

warehouse. There is no minimum volume required to keep a warehouse open.

1

150,000

2

2100,000

3

1

250,000

20,000

30,000

60,000

Figure 9.13. Potential Production-Distribution Network

Warehouse 1 has a fixed cost of $100,000, an inventory carrying cost of $0.5/cwt, a handling cost of

$2/cwt, and a handling capacity of 110,000 cwt. Warehouse 2 has a fixed cost of $500,000, an


inventory carrying cost of $1.5/cwt, a handling cost of $1/cwt, and a handling capacity of 500,000 cwt.

Plant 1 has a production capacity of 60,000 cwt and 50,000 cwt for products 1 and 2, respectively, and

production cost of $4/cwt and $3/cwt for products 1 and 2, respectively. Plant 2 has a production

capacity of 400,000 cwt and 500,000 for products 1 and 2, respectively, and production cost of $5/cwt

and $4/cwt for products 1 and 2, respectively. The demands for each product at the three customers are

shown in the Figure 13, where the solid lines indicated product 1 and the dashed lines indicated product

2. All material flows are expressed in hundredweight (cwt). The transportation costs per hundredweight

are given in the following table.

Table 9.3. Transportation Cost per Hundredweight (cwt)

Origin Destination Cost CostProduct 1 Product 2

P1 W1 0 0P1 W2 5 5P2 W1 4 4P2 W2 2 2W1 C1 4 3W1 C2 3 2W1 C3 5 4W2 C1 2 3W2 C2 1 2W2 C3 2 3

Determine the optimal production-distribution system using the mixed integer programming solver of

your choice. Show in an appendix to your report the formulation that you used, with all parameters and

costs as numerical values.

M1

M2

C1

C2

C3

W1

W2

Figure 9.14. Arc-Based Single Echelon Supply Chain Example Illustration


Figure 9.15. Supply Chain Ballou Example Transportation Data

Figure 9.16. Supply Chain Ballou Example Facility Capacity and Cost Data


Figure 9.17. Arc-Based Supply Chain Ballou Example Transportation Model


Figure 9.18. Arc-Based Supply Chain Ballou Example Facility Capacity Model

Figure 9.19. Arc-Based Supply Chain Ballou Example Solver


Figure 9.20. Arc-Based Supply Chain Ballou Example Transportation Solution


Figure 9.21. Arc-Based Supply Chain Ballou Example Facility Capacity Solution


Figure 9.22. Path-Based Supply Chain Ballou Example Transportation Model


Figure 9.23. Path-Based Supply Chain Ballou Example Facility Capacity Model

Figure 9.24. Path-Based Supply Chain Ballou Example Solver


Figure 9.25. Path-Based Supply Chain Ballou Example Transportation Solution


Figure 9.26. Path-Based Supply Chain Ballou Example Facility Capacity Solution

AMPL Model and Solution File # Data for the mathematical model for Strategic Network Design, prepared by # Edgar E. Blanco, Themis Rafailidis, Marcos Katsoulakis on June 2, 1997 #=========Set Definition ======= #--- commodities (index i) set COMMODITY:=C1 C2; #--- production plants set PLANT:=P1 P2; #--- warehouses set WAREHOUSE:=W1 W2; #--- customers set CUSTOMER:=CU1 CU2 CU3; #==========Constraint Paramenters ========== param capacity: P1 P2:= C1 60000 2000000 C2 50000 2000000; # Plant 2 has infinite capacity, but the total demand is never greater than 2000000 units! param demand: CU1 CU2 CU3:= C1 50000 100000 50000 C2 20000 30000 60000; param min_ware := W1 0 W2 0; param max_ware := W1 110000 W2 400001; #==========Cost Paramenters ========== param fix_ware:= W1 100000 W2 500000; param thr_ware:=


W1 2 W2 1; param carr_ware:= W1 0.5 W2 1.5; param unit_cost:= [C1,P1,*,*]: CU1 CU2 CU3:= W1 8 7 9 W2 11 10 11 [C2,P1,*,*]: CU1 CU2 CU3:= W1 6 5 7 W2 11 10 11 [C1,P2,*,*]: CU1 CU2 CU3:= W1 13 12 14 W2 9 8 9 [C2,P2,*,*]: CU1 CU2 CU3:= W1 11 10 12 W2 9 8 9; #----Constraint Parameters param capacity{COMMODITY, PLANT} >=0;

#supply (production capacity) for commodity i at plant j param demand{COMMODITY, CUSTOMER} >=0;

#demand for commodity i in demand zone l param min_ware{WAREHOUSE} >=0;

#minimum allowed annual throughput for the warehouse k param max_ware{WAREHOUSE} >=0;

#maximum allowed annual throughput for the warehouse k #---- Cost Parameters (annual based) param fix_ware{WAREHOUSE};

#fixed cost for opening and operating the warehouse k param thr_ware{WAREHOUSE};

#variable cost per unit of flow out of warehouse k param carr_ware{WAREHOUSE};

#inventory carrying cost per unit of flow out of warehouse k param unit_cost{COMMODITY,PLANT,WAREHOUSE,CUSTOMER}; # average unit cost of producing commodity i in plant j and # shipping it through warehouse k to customer l #----Variable Definition var flow{COMMODITY,PLANT,WAREHOUSE,CUSTOMER} >=0; # amount of units of commodity i produced in plant j and # shipped through warehouse k to customer l var serve_cust{WAREHOUSE,CUSTOMER} binary; # 1 if warehouse k serves customer l, 0 otherwise var open_ware{WAREHOUSE} binary; # 1 if warehous k is opened, 0 otherwise #----Model description #-----Objective Function minimize total_cost: sum{i in COMMODITY, j in PLANT, k in WAREHOUSE, l in CUSTOMER} unit_cost[i,j,k,l]*flow[i,j,k,l] + sum{k in WAREHOUSE} ( fix_ware[k]*open_ware[k] + (thr_ware[k]+carr_ware[k])*(sum{l in CUSTOMER} (sum{i in COMMODITY} demand[i,l])*serve_cust[k,l]) ); #----Constraints #------Available production capacity subject to Prod_Capacity{i in COMMODITY, j in PLANT}: sum{k in WAREHOUSE, l in CUSTOMER} flow[i,j,k,l] <= capacity[i,j]; #------All the demand must be met subject to Satisfy_Demand{i in COMMODITY, k in WAREHOUSE, l in CUSTOMER}: sum{j in PLANT} flow[i,j,k,l] = demand[i,l]*serve_cust[k,l]; #-----Single customer assignment to warehouse subject to Single_Customer{l in CUSTOMER}: sum{k in WAREHOUSE} serve_cust[k,l]=1; #-----Keeps warehouse throughput in the limits subject to Min_Flow{ k in WAREHOUSE}: sum{l in CUSTOMER} ( sum {i in COMMODITY} demand[i,l]) * serve_cust[k,l] >= min_ware[k]*open_ware[k];


subject to Max_Flow{ k in WAREHOUSE}: sum{l in CUSTOMER} ( sum {i in COMMODITY} demand[i,l]) * serve_cust[k,l] <= max_ware[k]*open_ware[k];

AIMMS Model and Solution File ! Strategic_Network_Design ! David J. Newton, Gina Robinson, Jeff Day ! May 28, 1997 ! file SND.aim SETS: Production_Plants, Warehouses, Customers, Products; INDICES: i in Production_Plants, j in Warehouses, k in Customers, p in products; PARAMETERS: carrying_costs(j) -> [0,inf] "indictes the inventory carrying costs for warehouse j", Production_costs(i,p) -> [0,inf] "the variable production costs per unit at plant i", Warehouse_capacity(j) -> [0,inf] "the maximum capacity constraint for warehouse j", Production_capacity(i,p) -> [0,inf] "the maximum capacity constraint for plant i", Warehouse_cost(j) -> [0,inf] "fixed cost of warehouse j being open ", Demand(k,p) -> [0,inf] "customer demands from warehouse j for product p", Transportation_cost(i,j,p) -> [0,inf] "cost to transport goods from plant i to warehouse j", Transportation_cost2(j,k,p) -> [0,inf] "cost to transport goods from warehouse j to customer k"; VARIABLES: x1(i,j,p) -> [0,inf] "flow between plant i and warehouse j", x2(j,k,p) -> [0,inf] "flow between warehouse j and customer k", y_customer(j,k) -> {0,1} "1 if customer k is being served by warehouse j - 0 otherwise", y_warehouse(j) -> {0,1} "1 if warehouse is to be open at location j - 0 otherwise", y_plant(i) -> {0,1} "1 if warehouse is to be open at location i - 0 otherwise", Total_Cost; CONSTRAINTS: Warehouse_Capacity_constraint(j).. Sum[(i,p), x1(i,j,p)] <= warehouse_capacity(j) * y_warehouse(j), Production_Capacity_constraint (i,p) .. Sum[(j), x1(i,j,p)] <= Production_capacity(i,p) * y_plant(i), Demand_Constraint (j,k,p) .. x2(j,k,p) = Demand(k,p)*y_customer(j,k), Balance_Constraint (j,p).. Sum[(i),x1(i,j,p)] = Sum[(k),x2(j,k,p)], Warehouse_Constraint(k).. Sum[(j), y_customer(j,k)] = 1, Total_Cost_Function .. Total_Cost = Sum[(i,j,p), x1(i,j,p)*(Production_costs(i,p)+Transportation_cost(i,j,p))] + Sum[(j), y_warehouse(j)*Warehouse_cost(j)] + Sum[(i,j,p), x1(i,j,p)*carrying_costs(j)] + Sum[(j,k,p), x2(j,k,p) * Transportation_cost2(j,k,p)]; MODEL: Strategic_Network_Design minimize : Total_Cost subject to: all method: mip;


! Data Section $ include a:SND.dat ! Execution Section SOLVE Strategic_Network_Design; display Total_Cost, x1, x2, y_warehouse, y_plant, y_customer; !file snd.dat SETS: Production_Plants := {M1, M2}, Warehouses := {W1, W2}, Customers := {C1, C2, C3}, Products := {P1, P2}; COMPOSITE TABLE: !Warehouse handling and carrying costs j carrying_costs !-------------------------------------- W1 2.5 W2 2.5; COMPOSITE TABLE: !Cost to produce each product at plant i i p production_costs !--------------------------------------------------- M1 P1 4 M1 P2 3 M2 P1 5 M2 P2 4; COMPOSITE TABLE: !Capacity of Warehouse j j Warehouse_capacity !-------------------------------------------------- W1 110000 W2 500000; COMPOSITE TABLE: !Production capacity of each product at plant i i p Production_capacity !-------------------------------------------------- M1 P1 60000 M1 P2 50000 M2 P1 200000 M2 P2 110000; COMPOSITE TABLE: !Table of fixed warehouse opening cost j Warehouse_cost !-------------------------------------------------- W1 100000 W2 500000; COMPOSITE TABLE: !Demand of each customer for each product k p Demand !--------------------------------------------------- C1 P1 50000 C1 P2 20000 C2 P1 100000 C2 P2 30000 C3 P1 50000 C3 P2 60000; COMPOSITE TABLE: !Transportation costs from plant to warehouse i j p Transportation_cost !------------------------------------------------------------------------- M1 W1 P1 0 M1 W2 P1 5 M2 W1 P1 4


M2 W2 P1 2 M1 W1 P2 0 M1 W2 P2 5 M2 W1 P2 4 M2 W2 P2 2; COMPOSITE TABLE: !Transportation costs from warehouse to customer j k p Transportation_cost2 !------------------------------------------------------------------------- W1 C1 P1 4 W2 C1 P1 2 W1 C2 P1 3 W2 C2 P1 1 W1 C3 P1 5 W2 C3 P1 2 W1 C1 P2 3 W2 C1 P2 3 W1 C2 P2 2 W2 C2 P2 2 W1 C3 P2 4 W2 C3 P2 3;

9.6. Fixed Costs Calculations One of the most difficult aspects of strategic distribution models is the determination of the proper fixed

cost for large capital-intensive assets such as facilities, buildings, and major machining lines. A further

complication is the interaction with corporate accounting and their rules and with the taxing authorities

and their rules. While in many operational and tactical decision support systems the impact of taxes can

be ignored, this is no longer the case for strategic decisions.

After-Tax Fixed Costs Calculations

Example

Assume a corporation has a constant marginal tax rate (mtr) of 40 %, a minimum attractive rate of return

(MARR) of 15 %. The company is considering the purchase of a major manufacturing asset with a

useful life (L) of seven years.

The first task is compute the net present value (NPV) of the cost and income flows associated with the

purchase of the machine. The purchase price or initial investment cost is $2,000,000, the useful life (L)

is seven years, there is no salvage value at the end of the seventh year, and the corporation is using a

straight-line depreciation. Let cft denote the cash flow at the end of period t, where the cash flow is

positive for revenue or income and is negative is expense or cost and time period zero denotes the

current time. The net present value is then computed as:


NPV cfMARR

tt

t

L=

+=∑ ( )10

(9.25)

NPV

NPV

=−

++

++

++

++

++

++

++

+= − + + + +

+ + + = −

20000001 015

2857141 015

2857141 015

2857141 015

2857141 015

2857141 015

2857141 015

2857141 015

2000000 248477 216041 187862163358 142050 123522 107411 811309

0 1 2

4 5 6

( . ) ( . ) ( . ) ( . )

( . ) ( . ) ( . ) ( . )

3

7

The Annual Equivalent Value (AEV) is then computed as follows:

AEV NPV A P MARR L= ⋅( / , , ) (9.26)

AEV = − ⋅ = −811 309 0 2404 195 039, . ,

It is the Annual Equivalent Value that is used as the fixed cost in strategic logistics systems design

models, provided the basic time period in the model is a year.

The capital recovery factor (crf) is obtained by the dividing the annual equivalent value by the initial

investment

crf =−

−=

1950392000000

00975,, ,

.

This capital recovery factor can now be used to obtain the after tax annual equivalent cost of any

equipment or initial investment, provided the assumptions about the minimum attractive rate of return,

the marginal tax rate, the investment life, the salvage value, and the depreciation schedule remain

unchanged. A more extensive treatment of engineering economics for capital projects can be found in

Park and Sharp (1990).

9.7. SMILE Integrated Distribution Model

Introduction The following integrated distribution design model has been developed by Wei and Goetschalckx. The

section comprises four subsections: verbal formulation, notation, detailed development, and

mathematical formulation. Much of the model was developed jointly with other researchers in the

Logistics Systems Program of the Material Handling Research Center, including M. Cole and K. Dogan.


Current Trends in Logistics Models and Algorithms

Computer Software Trends

• More capable MIP solvers

• Modeling languages

• ERP Systems (with some optimization)

Acceptance of Integrated Supply Chain View

• More comprehensive models (cradle to grave)

• More realistic models (inventory)

Models and Solution Algorithm Trends

• Models Growing in Complexity and Realism

• Models Developed by Supply Chain Owners

• Solution Techniques Become More Generic and “Shrink Wrapped”

Supply Chain Modeling and Algorithms Challenges

Multiple Periods

• Periodic demand

• Dynamic, multiperiod strategic systems

Global

• Taxes and profit realization

• Local contents, duty drawback

Stochastic

• Safety inventory

• Flexibility, robustness, risk, scenarios

Large Scale Models


Non-Linear Models

Stochastic Models

Standard MIP Linear Algorithms Cannot Solve Very Large Cases

NL-MIP or Stochastic Algorithms Only for Small Cases or Nonexistent

SMILE Production-Distribution Model

Model Characteristics

Multicommodity

Multi-echelon

Capacitated facilities

Capacitated channels

All costs

Deterministic

Single period

Single country or domestic

Arc formulation

Safety inventory proportional to demand (user determined)

Production cost and capacities

Weight and volume capacities

Depot and product single sourcing

Smart (tight) formulations

CPLEX MIP module


Verbal Formulation Minimize total cost =

warehouse inventory cost (Z1)

+ warehouse assembly cost (Z2)

+ warehouse facility cost (Z3)

+ warehouse variable cost (Z4)

+ plant facility cost (Z5)

+ plant production cost (Z6)

+ plant inventory cost (Z7)

+ shipment cost (Z8)

+ trunking inventory cost (Z9)

Subject to:

flow conservation (FC)

general flow conservation (GF)

one warehouse type per site (DT)

warehouse storage capacity (ST, IS)

warehouse flow capacity (TH, IF)

maximum/minimum open warehouses (XD, ID)

plant either open or closed (PO)

plant production capacity (SP)

minimum/maximum number of carriers (IC, XC)

carrier weight/volume capacity (WE, VO)

ship to/from open warehouse only (TD,FD)

ship from open plant only (FP)


customer single sourcing (SS1, 2, 3, 4)

max customer to warehouse distance (MD)

max customer to warehouse time (MT)

customer demand satisfaction (DM, SDM)

restrictions on variable values

Notation For consistency, all time-based parameters and variables are described in terms of days. Another

fundamental time period can be substituted everywhere for days.

Sets and Indexes

B set of manufacturing facilities or plants

C set of customers (B ∩ C = ∅)

CS set of customers with single sourcing requirements (CS ⊆ C)

D set of warehouses or depots (B ∩ D = ∅, C ∩ D = ∅)

L(j) set of warehouse types or sizes (e.g., small, medium, large, public) at warehouse site j (j ∈ D)

M(i,j) set of parallel transportation channels to represent different transportation modes or cost structures (e.g., truckload, less-than-truckload, carload) from facility i to facility j (i ∈ B ∪ D, j ∈ D ∪ C), possibly empty set

P set of products

PN set of products not involved in assembly operations (PN ⊆ P)

PC set of all products which can be assembly components for other products (PC ⊆ P)

PF set of all final products of all the assembly operations (PF ⊆ P)

PF(p) set of all final products which can include product p as an assembly component (PF(p) ⊆ PF)

i, j, k index for plants, customers, or warehouses

l index for warehouse types

m index for transportation modes

p index for products

f index for final products


Parameters

ProdnCapi production capacity of plant i (resource units per day)

PlantFixedCosti fixed cost of plant i ($ per day)

PlantClosingCosti cost to close plant i, 0 unless the plant is currently open ($ per day)

PlantProdnCostip production cost rate of plant i to produce one unit of product p ($ per unit of product p)

ProdnResUnitip production resource units required for plant i to produce one unit of product p (resource units per unit of product p)

WhseFixedCostjl fixed cost of type l warehouse at site j ($ per day)

WhseClosingCostj cost to close an existing warehose at site j

WhseFlowCostjlp flow cost of product p through type l warehouse at site j ($ per unit flow of product p)

WhseFlowCapjl maximum resource capacity available at type l warehouse at site j (resource units per day)

FlowResUnitjlp flow resource units required for type l warehouse at site j to handle one unit of product p (resource units per unit of product p)

WhseStgCostjl storage capacity cost of type l warehouse at site j ($ per volume storage capacity per day)

WhseStgCapjl maximum storage capacity of type l warehouse at site j (volume unit storage)

SSFactorjp safety stock factor of product p at warehouse j

AssyCostjf cost to assemble one unit of final product f at warehouse j ($ per unit product f)

AssyAmtpf units of component product p required to assemble one unit of final product f

MaxWhseOpen maximum allowable number of warehouses that can be open

MinWhseOpen manimum number of warehouses required to be open

Demandkp demand of customer k for product p, possibly zero (unit product f per day)

MaxDistancek maximum allowable distance from customer k to its server warehouse (miles)

MaxTimek maximum allowable travel time to serve a customer k from its server warehouse (days)

Distancejkm travel distance on transportation channel jkm (miles)

ijmTransitTime travel time on transportation channel ijm (planning periods). This time interval is expressed in planning periods and thus this number is likely to be much smaller than 1. For example, if the planning period is one year and the travel time is a day, then

. 0.0027397TransitTime =


ijmR time between shipments on transportation channel ijm (planning periods), i.e., order interval or replenishment interval. This time interval is expressed in planning periods and thus this number is likely to be much smaller than 1. For example, if the planning period is one year and the order interval is a week, then . 0.01923R =

ijmCarrierCost transport carrier cost on channel ijm ($ per carrier shipment)

ijmpTranUnitCost transportation variable cost per unit of product p of transportation channel ijm ($ per unit p shipment)

ijmMinCarriers minimum required number of carriers per order interval on transportation channel ijm if the channel is used (carriers per order interval)

ijmMaxCarriers maximum allowable number of carriers per order interval on transportation channel ijm if the channel is used (carriers per order interval)

ijmCarrierWtCap carrier weight capacity of transportation channel ijm (weight units per carrier)

ijmCarrierVolCap carrier volume capacity of transportation channel ijm (volume units per carrier)

Valuep value of a unit of product p ($ per unit of product p)

Wtp weight of a unit of product p (weight units per unit of product p)

Volp volume of a unit of product p (volume units per unit of product p)

r inventory carrying cost rate ($ per $ per day)

Decision Variables

ijmpx amount of product p shipped through transportation channel ijm (units of product p per planning period)

vjlp amount of product p flow through warehouse of type l at site j (units of product p per planning period)

ujlp maximum amount of product p stored at warehouse of type l at site j (units of product p)

ijmw number of carriers used per planning period from source facility i to destination facility j via transportation mode m (carriers per planning period)

si (0,1) 1 if plant is opened at site i, 0 otherwise

sci (0,1) 1 if plant is not opened at site i, 0 otherwise

yjl (0,1) 1 if type l warehouse is opened at site j, 0 otherwise

ycj (0,1) 1 if a warehouse is not opened at site j, 0 otherwise

zijm (0,1) 1 if transportation channel ijm is opened, 0 otherwise


q jk ∈ 0 1,l q 1 if customer k is served from facility j, 0 otherwise

Detailed Development This section contains a detailed mathematical development of the model. The notation is defined earlier

in the notation section. Recall that all costs, time-based parameters, and variables are amortized and

defined in terms of days to ensure that they are dimensionally consistent.

Products

The model can accommodate multiple products. Each product has a known weight, volume, and value

that do not change during the course of distribution. Some products may not be produced at a plant or

handled at a warehouse. If a plant or a warehouse can not process a product, the product flow from that

plant or warehouse will be zero. For simplification, details are omitted from the model. In creating the

formulation, e.g., the MPS file, it is easy to preprocess out those variables.

Warehousing

This section includes development of costs and constraints associated with warehouses.

Warehousing Inventory

This section develops the warehouse inventory cost function. It is assumed that every R days, each

warehouse places an order for a product. The sketch below shows an idealized single product inventory

cycle.

Time

Inventory

Q

SS

0 t t+R

Figure 9.27. Idealized Single Product Inventory Cycle


Warehouse j placed an order and after the lead-time the order is received at time t. The inventory reach

its highest level Q (Q = SS + demand rate × R) at time t. The cycle repeats after the next order is placed.

The next order will be received at time t+R. Since the order interval is assumed to be the same as the

shipping interval of the transportation channel from that source facility supplying the warehouse, R

depends on the supply channel selected.

The warehouse inventory cost is simply the product of the inventory holding cost rate and the value of

the average inventory. The mathematical form is developed below.

DemandRatejp = demand rate for product p at warehouse j

Qjp = expected maximum inventory of product p at warehouse j

SSjp = safety stock of product p at warehouse j

Average Inventoryjp = average inventory of product p at warehouse j

DemandRate = jpi B D m M i j

ijmpx∈ ∪ ∈∑ ∑

( , )

For convenience of modeling, it's assumed that the worst possible case in which all product orders are

replenished simultaneously.

Q SS Cycle Inventoryjp jp jp= + _

Cycle Inventory xjp ijm ijmpm M i ji B D

_( , )

= R∈∈ ∪∑∑

Safety stock is the product flow times a preset safety stock factor, e.g., three days of product flow.

SS = SSFactor jp jp ijmpm M i ji B D

x∈∈ ∪∑∑

( , )

Q = SSFactor Rjp jp ijm ijmpm M i ji B D

x+∈∈ ∪∑∑ d i

( , )

Average_ Inventory xjp jpijm

ijmpm M i ji B D

= SSFactorR

+FHG

IKJ∈∈ ∪

∑∑ 2( , )

Thus, the warehouse inventory cost for warehouse j is

r Value SSFactorR

p jpijm

ijmpm M i ji B Dp P

x+FHG

IKJ∈∈ ∪∈

∑∑∑ 2( , )


and the total system wide warehouse inventory cost is

r Value SSFactorR

p jpijm

ijmpm M i ji B Dp Pj D

x+FHG

IKJ∈∈ ∪∈∈

∑∑∑∑ 2( , ) (Z1)

Warehouse Assembly Operations

This section develops the model of the assembly operations at warehouses. Moiseenko (1984) presents

a generalized multicommodity network flow model which allows conversion of different commodities.

Still, commodity conversion can not model assembly operations; the product assembly is a more general

problem.

For those products that are not involved in any assembly operations, constraint (FC) enforces

conservation of flow at each warehouse.

i B D m M i j

ijmpk C D m M j k

jkmpx∈ ∪ ∈ ∈ ∪ ∈∑ ∑ ∑ ∑=

( , ) ( , )x j D p PN∈ ∈, (FC)

However, flow conservation does not hold true for assembly operations. Consider the following

example where two units of product A and three units of product B are combined to form one unit of

product C and two units of product B and one unit of product D are combined to form one unit of

product E. The typical assembly equations for the operations are:

2 A + 3 B → C

2 B + 1 D → E

In this case, the flow conservation condition must be replaced by some more general constraints. Let x

denote the input flows and y denote the output flows. The general flow conservation equations for the

warehouse assembly operations are

x y y x

x y y x y xx y y x

A A C C

B B C C E E

D D E E

− = −

− = − + −

− = −

2

3 2

b gb g b g

The general flow conservation conditions for warehouse assembly operations are as follows:

AssyAmt

i B Dm M i jijmp

k C Dm M j kjkmp

pfm M j k

jkmfk C D m M i j

ijmfi B Df PF p

x x

x x

∈ ∪ ∈ ∈ ∪ ∈

∈∈ ∪ ∈∈ ∪

F

HGG

I

KJJ

∈

∑ ∑ ∑ ∑

∑∑ ∑∑∑

− =

−

( , ) ( , )

( , ) ( , )( )

j D p PC∈ ∈, (GF)


The total warehouse assembly cost is

AssyCost jfk C D m M j k

jkmfi B D m M i j

ijmff PFj D

x∈ ∪ ∈

−∈ ∪ ∈

xF

HGG

I

KJJ

∈∈∑ ∑ ∑ ∑∑∑

( , ) ( , ) (Z2)

Warehouse Facility

The facility cost of each warehouse includes a cost proportional to storage capacity, a cost proportional

to the amount shipped through the warehouse, and a fixed cost to either operate a warehouse or to close

a warehouse. The fixed cost to operate a warehouse accounts for overhead, capital costs, and other costs

that are not considered to be proportional to storage capacity or throughput. The cost to close an

existing warehouse accounts for the various expenditures needed to cease operations at a particular site.

Each of these costs is defined in terms of a warehouse site and a warehouse type. In general, each

potential warehouse site can be occupied by one of several types of warehouses. For example, small,

medium, and large may be three possible types of warehouse. The notion of multiple warehouse types,

each with a linear cost structure, allows a nonlinear (but piecewise linear) cost structure for each

potential warehouse site.

Each potential warehouse either be opened to exactly one type or be closed.

( )

1jl jl L j

y yc j∈

+ = ∈∑ D (DT)

The warehouse fixed and closing costs are expressed in (Z3)

WhseFixedCost + WhseClosingCostjl jll L j

j jj D

y yc∈∈∑∑FHGG

IKJJ( )

(Z3)

All products are assumed to be handled and stored in the same volume units, e.g., pallets, boxes, or

trucks. All the warehouse capacities, warehouse storage costs, and warehouse handling costs are

expressed in terms of the same volume units.

The worst case maximum amount of product p stored at warehouse of type l at site j, ujlp, is defined by

the constraints (ST) and (IS). The right hand side of constraint (IS), the worst case maximum inventory

at the warehouse, is developed above in the section about warehouse inventory.

Constraint (ST) enforces the required warehouse storage capacity for each warehouse type.


Vol WhseStgCap p jlpp P

jl jlu∈∑ ≤ y j D l L j∈ ∈, ( ) (ST)

The amount of product p shipped through warehouse of type l at site j, vjlp, is defined by the constraints

(TH) and (IF). Constraint (IF) states that the amount shipped through a warehouse is equal to the

product flow exists a warehouse.

v jlpl L j k C D m M j k

jkmp∈ ∈ ∪ ∈∑ ∑ ∑=

( ) ( , ) x j D p P∈ ∈, (IF)

Constraint (TH) enforces that the flow through a warehouse cannot exceed the maximum handling

capacity for each warehouse type. This constraint can also be used to model a general resource

consumption proportional to the flow and resource capacity constraint.

FlowResUnit WhseFlowCap jlp jlpp P

jl jlv∈∑ ≤ y j D l L j∈ ∈, ( ) (TH)

The warehouse variable operating costs are expressed in (Z4). The second cost component is most of

the times the warehouse handling cost, but can be any resource and cost proportional to the flow.

WhseStgCost Vol + WhseFlowCost jl pp P

jlp jlp jlpp Pl L jj D

u∈ ∈∈∈∑ ∑∑∑ v

FHGG

IKJJ( )

(Z4)

There will be at most specified maximum allowable open warehouses and at least specified minimum

required open warehouses.

Production

Plant costs include facility costs, production costs, and inventory costs. Plant facility costs comprise

fixed costs to either operate a plant or to close a plant.

PlantFixedCost + PlantClosingCosti i i ii B

sb∈∑ sc g (Z5)

The following constraints enforce that each plant must be either open or closed.

s sci i+ = 1 i B∈ (PO)

For each product, each plant is assumed to produce at a steady daily rate equal to the average number of

units it ships per day. Plant production costs are incurred proportional to the number of units a product


shipped out of a plant. The production cost per unit of product can differ between products and between

plants.

PlantProdnCostip ijmpp Pm M i jj C Di B

x∈∈∈ ∪∈∑∑∑∑

( , ) (Z6)

Plant inventory costs are incurred only on finished goods inventories. Finished stocks are assumed to be

accumulated at a steady rate. For each product and destination depot, the finished good inventory level

at a plant follows the familiar saw tooth pattern. The total plant inventory cost is

r Value R2FHGIKJ∈∈∈ ∪∈

∑∑∑∑ pp Pm M i j

ijm ijmpj C Di B

x( , )

(Z7)

Plant production capacity is expressed in terms of Production Resource Units. Constraint (SP) prevents

a plant from shipping products at a daily rate greater than its production capacity.

ProdnResUnits ProdnCap ip ijmp i ip Pm M i jj C D

x ≤∈∈∈ ∪∑∑∑

( , )s i B∈ (SP)

Transportation

The transportation between facilities, or trunking, and local delivery to the final customers are assumed

to be executed by direct shipment and vehicle routing is not considered. If routing cost estimates are

available, then the local delivery costs can be based on these estimates. The total amount of product

shipped during a planning period is assumed to be large compared to the size of a single carrier. Thus,

integrality effects related to the number of carriers used can be ignored. Shipment costs comprise a

variable cost proportional to the number of carriers used, the amount of weight units shipped, or both.

( , )

CarrierCost TranUnitCostijm ijm ijmp ijmpi B D j C D m M i j p P

w∈ ∪ ∈ ∪ ∈ ∈

+

∑ ∑ ∑ ∑ x (9.Z8)

Each unit shipped over channel ijm spends T time in-transit from i to j. Inventory costs

incurred in local delivery are ignored. Thus, the total transportation inventory costs are

ijmransitTime

( , )

p ijm ijmpi B D j D m M i j p P

r Value TransitTime x∈ ∪ ∈ ∈ ∈∑ ∑ ∑ ∑ (9.Z9)


Each transportation channel has a required minimum number of carriers used per order interval on that

transportation channel if the transportation channel is used. The minimum number, which may be zero,

is the enforced by constraint (IC).

MinCarriers , , ( , )ijm ijm ijm ijmR w z i B D j C D m≥ ∈ ∪ ∈ ∪ M i j∈ (9.IC)

Each transportation channel has a limited maximum allowable number of carriers used per order

interval. The maximum number, which may be infinity, is enforced by constraint (XC).

MaxCarriers , , ( , )ijm ijm ijm ijmR w z i B D j C D m≤ ∈ ∪ ∈ ∪ M i j∈ (9.XC)

Each carrier has a weight capacity, enforced by constraint (WE),

Wt CarrierWtCap , , ( , )p ijmp ijm ijmp P

x w i B D j C D m M i∈

≤ ∈ ∪ ∈ ∪∑ j∈ (9.WE)

Each carrier has a volume capacity, enforced by constraint (VO).

Vol CarrierVolCap , , ( , )p ijmp ijm ijmp P

x w i B D j C D m M i j∈

≤ ∈ ∪ ∈ ∪∑ ∈

j k

j∈

(9.VO)

Constraints (TD) and (FD) require that transportation channels to and from a potential depot site are

usable only if a depot is actually opened at that site.

( )

, , ( , )ijm jll L j

z y i B D j D m M i j∈

≤ ∈ ∪ ∈ ∈∑ (9.TD)

( )

, , ( , )jkm jll L j

z y j D k C D m M∈

≤ ∈ ∈ ∪ ∈∑ (9.FD)

Constraint (FP) requires that transportation channels from a plant are usable only if a plant is open.

, , ( , )ijm iz s i B j C D m M i≤ ∈ ∈ ∪ (9.FP)

Constraints (SS1) enforces that the customers requiring single sourcing be served by exactly one source.

q jkj B D∈ ∪

∑ = 1 k CS∈ (9.SS1)

Constraints (SS2 and SS3) require that a customer can only be served from an open plant or an open

warehouse.

q sik i≤ i B k CS∈ ∈, (9.SS2)


q yjk jll L j

≤∈∑

( )j D k CS∈ ∈, (9.SS3)

Constraint (SS4) requires that transportation channels from a source to a customer are usable only if the

customer is served by that source.

, , ( , )jkm jkz q j B D k CS m M j≤ ∈ ∪ ∈ ∈ k (SS4)

Constraints (MD) and (MT) require that each customer be within a specified maximum travel distance

and time from the warehouse or warehouse that serves it.

( )Distance MaxDistance , , ,jkm jkm kz k C j B≤ ∈ ∈ ∪ D m M j k∈ (MD)

( )TransitTime MaxTime , , ,jkm jkm kz k C j B D≤ ∈ ∈ ∪ m M j k∈ (MT)

Customers

Constraint (DM) ensures demand satisfaction. Constraints (DM) are for customers without single

sourcing requirements.

( , )

=Demand , ,jkmp kpj B D m M j k

x k C k CS p P∈ ∪ ∈

∈ ∉ ∈∑ ∑ (DM)

Constraints (SDM) are for customers with single sourcing requirements.

( , )

Demand , ,jkmp jk kpm M j k

x q j B D k∈

= ∈ ∪∑ CS p P∈ ∈

( , )

(SDM)

Other customer-related constraints involve transportation; they are listed in section 4.3.3.

Decision Variable Values

The values obtained by the decision variables are constrained as follows.

v jlp ≥ 0 j D l L j p P∈ ∈ ∈, ( ),

u jlp ≥ 0 j D l L j p P∈ ∈ ∈, ( ),

xijmp ≥ 0 i B D j C D m M i j p P∈ ∪ ∈ ∪ ∈ ∈, , ( , ),

wijm ≥ 0 i B D j C D m M i j∈ ∪ ∈ ∪ ∈, ,


si ∈ 0 1,l q i B∈

sci ∈ 0 1,l q i B∈

y jl ∈ 0 1,l q j D l L j∈ ∈, ( )

yc j ∈ 0 1,l q j D∈

zijm ∈ 0 1,l q i B D j C D m M i j∈ ∪ ∈ ∪ ∈, , ( , )

q jk ∈ 0 1,l q j B D k CS∈ ∪ ∈,


Mathematical Formulation Minimize

r Value SSFactorR

p jpijm

ijmpm M i ji B Dp Pj D

x+FHG

IKJ∈∈ ∪∈∈

∑∑∑∑ 2( , ) (Z1)

AssyCost jfk C D m M j k

jkmfi B D m M i j

ijmff PFj D

x∈ ∪ ∈

−∈ ∪ ∈

xF

HGG

I

KJJ

∈∈∑ ∑ ∑ ∑∑∑

( , ) ( , ) (Z2)

WhseFixedCost + WhseClosingCostjl jll L j

j jj D

y yc∈∈∑∑FHGG

IKJJ( )

(Z3)

WhseStgCost Vol + WhseFlowCost jl pp P

jlp jlp jlpp Pl L jj D

u∈ ∈∈∈∑ ∑∑∑ v

FHGG

IKJJ( )

(Z4)

PlantFixedCost + PlantClosingCosti i i ii B

sb∈∑ sc g (Z5)

PlantProdnCostip ijmpp Pm M i jj C Di B

x∈∈∈ ∪∈∑∑∑∑

( , ) (Z6)

r Value R2FHGIKJ∈∈∈ ∪∈

∑∑∑∑ pp Pm M i j

ijm ijmpj C Di B

x( , )

(Z7)

CarrierCostR

TranUnitCostijm

ijmijm ijmp ijmp

p Pm M i jj C Di B Dw x

FHG

IKJ +

LNMM

OQPP∈∈∈ ∪∈ ∪

∑∑∑∑( , )

(9.Z8)

r Value TransitTimep ijm ijmpp Pm M i jj Di B D

x∈∈∈∈ ∪∑∑∑∑

( , ) (Z9)

Subject to

i B D m M i j

ijmpk C D m M j k

jkmpx∈ ∪ ∈ ∈ ∪ ∈∑ ∑ ∑ ∑=

( , ) ( , )x j D p PN∈ ∈, (FC)

AssyAmt

i B Dm M i jijmp

k C Dm M j kjkmp

pfm M j k

jkmfk C D m M i j

ijmfi B Df PF p

x x

x x

∈ ∪ ∈ ∈ ∪ ∈

∈∈ ∪ ∈∈ ∪

F

HGG

I

KJJ

∈

∑ ∑ ∑ ∑

∑∑ ∑∑∑

− =

−

( , ) ( , )

( , ) ( , )( )

j D p PC∈ ∈, (GF)


y ycjll L j

j∈∑ + =

( )1 j D∈ (DT)

u jlpl L j

jp ijm ijmpm M i ji B D∈ ∈∈ ∪

∑ ∑∑= +( ) ( , )

SSFactor Rd ix j D p P∈ ∈, (IS)

Vol WhseStgCap p jlpp P

jl jlu∈∑ ≤ y j D l L j∈ ∈, ( ) (ST)

v jlpl L j k C D m M j k

jkmp∈ ∈ ∪ ∈∑ ∑ ∑=

( ) ( , ) x j D p P∈ ∈, (IF)

FlowResUnit WhseFlowCap jlp jlpp P

jl jlv∈∑ ≤ y j D l L j∈ ∈, ( ) (TH)

y jll L jj D ∈∈∑∑ ≤

( )MaxWhseOpen (XD)

y jll L jj D ∈∈∑∑ ≥

( )MinWhseOpen (ID)

1i is sc i B+ = ∈ (PO)

( , )

ProdnResUnits ProdnCapip ijmp i ij C D m M i j p P

x s∈ ∪ ∈ ∈

≤∑ ∑ ∑ i B∈

M i j∈

M i j∈

(SP)

MinCarriers , , ( , )ijm ijm ijmw z i B D j C D m≥ ∈ ∪ ∈ ∪ (IC)

MaxCarriers , , ( , )ijm ijm ijmw z i B D j C D m≤ ∈ ∪ ∈ ∪ (XC)

R Wt CarrierWtCap , , ( , )ijm p ijmp ijm ijmp P

x w i B D j C D m∈

≤ ∈ ∪ ∈∑ M i j∪ ∈ (WE)

R Vol CarrierVolCap , , ( , )ijm p ijmp ijm ijmp P

x w i B D j C D m∈

≤ ∈ ∪ ∈∑ M i j∪ ∈

j k

j∈

k CS∈

(VO)

( )

, , ( , )ijm jll L j

z y i B D j D m M i j∈

≤ ∈ ∪ ∈ ∈∑ (TD)

( )

, , ( , )jkm jll L j

z y j D k C D m M∈

≤ ∈ ∈ ∪ ∈∑ (FD)

, , ( , )ijm iz s i B j C D m M i≤ ∈ ∈ ∪ (FP)

1jkj B D

q∈ ∪

=∑ (SS1)


, ik iq s i B k C≤ ∈ S∈

D k CS∈

k

CS m M j k∈

(SS2)

( )

, jk jll L j

q jy∈

≤ ∈∑ (SS3)

, , ( , )jkm jkz q j B D k CS m M j≤ ∈ ∪ ∈ ∈ (SS4)

Distance MaxDistance , , ( , )jkm jkm kz j B D k≤ ∈ ∪ ∈ (MD)

( )TransitTime MaxTime , , ,jkm jkm kz k C j B D≤ ∈ ∈ ∪ m M j k∈ (MT)

( , )

=Demand , ,jkmp kpj B D m M j k

x k C k CS p P∈ ∪ ∈

∈ ∉ ∈∑ ∑ (DM)

( , )

Demand , ,jkmp jk kpm M j k

x q j B D∈

= ∈ ∪∑ k CS p P∈ ∈ (SDM)

v jlp ≥ 0 j D l L j p P∈ ∈ ∈, ( ),

u jlp ≥ 0 j D l L j p P∈ ∈ ∈, ( ),

xijmp ≥ 0 i B D j C D m M i j p P∈ ∪ ∈ ∪ ∈ ∈, , ( , ),

wijm ≥ 0i B D j C D m M i j∈ ∪ ∈ ∪ ∈, , ( , )

{ }si ∈ 0 1, i B∈

{ }sci ∈ 0 1, i B∈

{ }y jl ∈ 0 1, j D l L j∈ ∈, ( )

{ }yc j ∈ 0 1, j D∈

{ }zijm ∈ 0 1, i B D j C D m M i j∈ ∪ ∈ ∪ ∈, , ( , )

{ }q jk ∈ 0 1, j B D k CS∈ ∪ ∈,

The above model belongs to the class of large scale mixed integer programming formulations. These

models have been much more difficult to solve than linear or network flow programming problems of

equivalent size.


Optimal Solution Algorithms

Preprocessing

Preprocessing is a technique that simplifies rows and columns by eliminating redundant rows and

variables, substituting and fixing variables, and increasing lower and decreasing upper bounds on

variables. Some general MIP preprocessing techniques are discussed in Savelsbergh (1992) and have

been implemented in MINTO. Some general LP preprocessing techniques are have been implemented

in CPLEX. The preprocessing approach presented here uses the logistics problem information that can

be easily identified and implemented from the problem data.

For several of the constraints, coefficients and right-hand sides can be tightened with little effort. Define

C(j) as the set of customers which can be reached from facility (plant or warehouse) j, either directly or

indirectly. Since the throughput of a facility can not exceed the total demand of all customers the

facility can reach, constraint (SP) and (TH) can be adjusted as follows.

PPR x PPR DM PPCap sip ijmpp Pm Mj D C

ipk C i

kpp P

i i∈∈∈ ∪ ∈∈∑∑∑ ∑∑≤

min ,

( )i B∈ (SP)

WHR v WHR DM WHCap yjlp jlp jlp kp jl jlp P

≤

∑∑∑

∈min , j D l L∈ ∈, (TH)

Constraints (MD) an (MT) can be processed out by eliminating those channels that violate the maximum

travel distance limit and the maximum travel time limit. Many cost components and constraints are

optional depending on the problem instance. These can be ignored without affecting the solution. For

example, all variables and constraints related to single sourcing, ( ) and (SS1)-(SS4), respectively, can

be eliminated if a global status variable indicates that no single sourcing is required.

qik

Valid Inequalities

For a particular product, the flow from a facility to a customer, who is directly supplied by this facility,

can not exceed the demand of that customer, The following first two sets of valid inequalities can then

be added to the constraint set to tighten the LP relaxation, corresponding to the cases where the

supplying facility is a depot or a plant, respectively. For a particular product and if a channel requires a

minimum number of carriers when the channel is used, then the flow for a particular product on that


channel cannot exceed the demand of that customer. The third set of valid inequalities (VI3) can then be

added to the constraint set to tighten the LP relaxation and it is valid for both plants and depots directly

supplying that customer.

, ,jkmp kp jlm M l L

x Demand y j D k C p P∈ ∈

≤∑ ∑ ∈ ∈ ∈ (VI1)

, ,ikmp kp im M

x Demand s i B k C p P∈

≤∑ ∈ ∈ ∈ (VI2)

, , ,ikmp kp ikmx Demand z i B D k C m M p P≤ ∈ ∪ ∈ ∈ ∈ (VI3)

These inequalities are very similar to the Strong Linear Programming Relaxation (SLPR) of the

Uncapacitated Facility Location problem (UFL). However, because the SLPR has a very large number

of constraints, it is not trivial to solve SLPR even once for the root node of the branch-and-bound tree.

We have found that just a small fraction of the total number of (VI1) and (VI2) constraints contributes to

a tighter relaxation. Since adding the (VI1) and (VI2) constraints will force the depot or plant open, it

seems logical to add those constraints corresponding to the customer demands that are more important

than others for a particular depot or plant. We selected the customers which are the closest to the depot

or plant and the customers that have the largest demand expressed in weight units. Also since the

transportation cost often is weight based, we selected first the (VI3) constraints for which the customer

product demands have higher total weight.

Adding only a fraction of the total number of valid inequalities has proven to be an efficient solution

strategy. Adding all valid inequalities created a linear relaxation that could not be solved in several

instances because of memory constraints or solving such a large linear relaxation required excessive

computer times. On the other hand, a branch-and-cut solution strategy successively generated valid but

irrelevant inequalities and its total solution time also exceeded the time required by adding a fraction of

the inequalities at the root node.

Branching Order

Finally, there exists a natural hierarchy in the design decisions for configuring production-distribution

networks. The decision involving the opening or closing of plants have the most far reaching impact.

Next, are the decisions regarding the establishment of a distribution center at a particular site. Once a

site has been selected, then the best type of distribution center has be chosen. Finally, after all facility


decisions have been made, then the transportation channels are selected and the (integer) number of

carriers is determined for each channel. This hierarchy is given to the mixed integer programming

solver in the form of a variable branching order. Again, the impact of selecting the wrong branching

order can be very large, as illustrated in the RECYCLE case study.

Case Studies The CIMPEL model and design environment were developed based on numerous real world case

problems. The performance of CIMPEL in sample of these cases will be briefly reviewed.

Electronics Manufacturing (ELEC)

The design problem was to decide the number, location and size of distribution centers for an electronics

manufacturer situated in the eastern and central section of the United States. There were 30

manufacturing plants, 30 different commodities, 98 customer facilities, 50 potential warehousing sites

each with 6 different types, 6400 transportation channels each with a single transportation mode. All

thirty products have weights equal to one and volumes equal to one. Each product can only be produced

at one specific plant and that plant can only produce that product.. Plant production is uncapacitated and

production cost was not modeled. Each warehouse has inbound channels from the plants and outbound

channels to customers. There are no channels between warehouses and no channels between plants and

customers, i.e., the distribution system consists of exactly one level of warehousing. All potential

warehouse sites are identical, except in their location. Each warehouse site can have one of six different

types, each with a fixed cost, handling cost, and handling capacity. The largest warehouse type can

accommodate all the customer demands. Those six types represent a concave piecewise linear cost

structure for each potential warehouse site. Warehouse storage is uncapacitated and storage cost was not

modeled. There is no closing cost and no inventory cost associated with warehousing. Transportation

cost is on a per unit basis. No customer requires single sourcing. Every customer requires the

warehouse that serves it to be within 1300 miles travel distance. Customer demands are deterministic

and exhibit a very wide range for different customers and products. The optimal configuration has only

one potential warehouse open to largest type which serves all the customers. The solution times for the

different solution methods are given in the next table.


Table 9.4. Solution Statistics for the ELEC Case

Problem Solved # Variables # Binary # Constraints Solution # of Nodes LP Relax. GapVars. Time (s) Evaluated 100(Z -ZLP)/Z %

Original No 119,858 350 5,555 > 20,072.66 1,482 15.48%Strong LP Relaxation No 119,858 350 123,563 > 54.76 0 Unknown10% Strong LP Relax. Yes 119,858 350 25,428 2413.91 0 0.00%

Clearly the original linear programming relaxation was to weak, so too many nodes had to be evaluated.

On the other hand, the strong LP relaxation created such a large linear program that it could not be

solved in the memory available (256 megabytes of core RAM). The LP relaxation with 10 % of the

strong constraints could however solve the problem in the allotted time and memory.

The effect of tightening constraint (TH) was tested on a test case which is a variation of ELEC with 25

potential warehouses, 55 customers, and all other characteristics remain unchanged. The case was

solved using CPLEX 2.1 on BERNINI (Pentium 90 PC). The solution times for the different solution

methods are given in the next table.

Table 9.5. Solution Statistics for the ELEC Case with Preprocessing

Solution Time Number of Nodes LP Relaxation Gap(seconds) Evaluated (Z -ZLP)/Z (%)

Before Tightening 5177.81 483 24.17After Tightening 838.56 194 8.79

Wholesaler (HAC)

The design problem was to decide the number, location and size of distribution centers for a wholesale

distributor of heating and air conditioning products. There were 18 manufacturing plants, 18 different

commodities, 242 customer facilities, 21 potential warehousing sites each with one type, 5460

transportation channels each with a single transportation mode. Plant production is uncapacitated and

production cost was not modeled. Each warehouse has inbound channels from the plants and outbound

channels to customers. There are no channels between warehouses and no channels between plants and

customers, i.e., the distribution system consists of exactly one level of warehousing. Potential

warehouse sites have different costs. Warehouse storage is uncapacitated and storage cost was not

modeled. There is no closing cost and no inventory cost associated with warehousing. Transportation

cost is on a per unit basis. No customer requires single sourcing. Customer demands are deterministic

and exhibit a very wide range for different customers and products. The optimal configuration has eight


potential warehouse open. The solution times for the different solution methods are given in the next

table.

Table 9.6. Solution Statistics for the HAC Case

Problem Solved # Variables # Binary # Constraints Solution # of Nodes LP Relax. GapVars. Time (s) Evaluated 100(Z -ZLP)/Z %

Original No 83,559 42 4,739 > 13,765 2,285 37.38%Strong LP Relaxation Yes 83,559 42 82,878 485 0 0.00%10% Strong LP Relax. Yes 83,559 42 21,665 192 23 0.24%

Clearly the original linear programming relaxation was to weak, so too many nodes had to be evaluated.

On the other hand, the strong LP relaxation created such a large LP that it took longer to solve the

overall problem even though the problem was solved at the root node. The LP relaxation with 20 % of

the strong constraints could however solve the problem in a smaller amount of time. Observe that with

one fourth of the total number of constraints the gap at the root node was less than a quarter of a percent.

Recycling System (RECYCLE)

The problem was to determine the location, number, and size of material recovery facilities in a three

county area for the collection and processing of recyclable materials. The four materials in this problem

were the materials most often collected during residential curb side collection: newspaper, glass,

aluminum, and plastic. These materials have to be sorted, potentially cleaned, and baled for final

transportation to various reclamation facilities. Since this is a recycling case, the real materials flow

from the customers, through the material recovery facilities, to the reclamation plants. In the model all

flows were still assumed to flow from plants, through depots, to customers. There were 5 reclamation

plants, 4 commodities, 9 customers, 3 material recovery facilities, each with three different sizes, and 48

transportation channels. There was only one transportation mode, the collection trucks. There are no

channels between warehouses and no channels between plants and customers, i.e., the distribution

system consists of exactly one level of warehousing. Potential warehouse sites have different costs.

Warehouse storage is uncapacitated and storage cost was not modeled. There is no closing cost and no

inventory cost associated with warehousing. . No customer requires single sourcing. Customer demands

are deterministic and exhibit a very wide range for different customers and products.

Because the collection trucks were very underutilized, a linear approximation of the number of carriers

on a transportation channel was no longer valid, i.e., the number of carriers is required to be an integer

number. The following branch order is used: first the solver branches on the depot binary variables and


then on the integer carrier variables. The solution times for the different solution methods are given in

the next table.

Table 9.7. Solution Statistics for the RECYCLE Case

Problem Solved # Variables # Binary # Constraints Solution # of Nodes LP Relax. Gap/ Int. Vars. Time (s) Evaluated 100(Z -ZLP)/Z %

Original No 273 12 & 45 209 > 55,140 333,300 76.49%Braching Order Yes 273 12 & 45 209 2.63 147 76.49%

Packaging Operations (PACKAGE)

This problem involved the strategic planning of the whole supply chain of a packaging company. In a

packaging operation paper roll stock from the paper mill is transformed into finished goods in two major

stages of manufacturing, called plants and finishing facilities, for which the machines are called presses

and gluers, respectively. The decisions included the supply mixes of the paper mills, allocation of the

manufacturing resources (i.e. presses and gluers) to major product groups, and the configuration of the

distribution system. The system included 2 paper mills, 9 potential locations for manufacturing and

warehousing operations, 24 types of machines to conduct these manufacturing operations, and 250

major customer locations. The machines and the facilities each had a fixed cost and a variable

production or handling cost. The machines, and some of the warehousing facilities had capacities. The

speed of machines varied depending on the product and the location. The customers had known

demands. The customer demands had a strong seasonal pattern. In an initial and traditional approach,

we conducted the study for the peak demand period.


C u s t o m e r s

Finishing Facility

P l a n t

M i l l

W a r e h o u s e

Facility

Presses

Facility

Gluers

Plant

F in i s h in g F a c i l i t i e s

Fixed Cost

Fixed & Var. Cost

Fixed Cost

Fixed & Var. Cost

Figure 9.28. Schematic of a Two Stage Manufacturing System including Machines

The model structure allowed us to represent each of the machines, be it either a press or a gluer, as a

distribution facility, while the facilities that housed the manufacturing resources also were modeled as

warehouses. Transportation channels assured that no machine was used unless the facility that housed

this machine was also used. The model described above was flexible and generic enough so that it could

represent this production and distribution network including the machines inside the facilities for which

was not initially designed. In the preceding figure the supply chain of the company is illustrated. The

implementation of 100% strong formulation allowed the system to be solved to optimality.

Table 9.8. Solution Statistics for the PACKAGE Case

Problem Solved # of # of Binary # of Solution Time # of Nodes LP Relax. GapVariables Variables Constraints (seconds) evaluated 100*(Z-ZLP)/Z %

100% Strong Yes 6446 317 4279 251 591 0.05%

Clearly, the fine tuning of the mixed integer programming formulation and solution algorithms has

reduced dramatically the time required to solve the problems and has extended the size of the problems

can be solved.

9.8. Conclusions Without doubt, there will be design problems where our model is not sufficiently accurate or detailed. A

prominent example is our assumption that the production cost and resource requirements for producing a

particular product at a manufacturing plant are independent of the production quantity or the product


mix produced at this plant. Also, our treatment of safety inventory is very simplistic but realistic.

International considerations, such as local content, duties, and taxes are also not currently included in

our system. At the same time, the execution times of our algorithms will be larger than that of

specialized models and solution techniques when designing systems of comparable size.

But, it was our goal to reduce the time required to design a production and distribution system for a

company without much prior experience in this area. The predefined structure of the model allows the

company to focus on the data collection and sensitivity analysis parts of the design task. The algorithm

enhancements allow SMILE to be solved for these real life cases in a fraction of the time of a general

purpose solver, if the general purpose solver can reach a solution at all. At the same time, the CIMPEL

modeling environment has been of significant help in communicating our models and algorithms with

industrial companies and with students in our undergraduate, graduate and continuing education classes.

We therefore believe there is a place for generic but domain specific design models, algorithms, and

modeling environments that bridge the gap between multipurpose algebraic modeling languages and

one-of-a-kind specialized models and associated solution algorithms.

Exercises

True-False Questions

An advantage of the path-based formulation over the arc-based formulation is that addition an additional

echelon between source and sink does not dramatically increase the number of required variables,

(T/F) _____(1).

For the design of strategic logistics systems the site generating algorithms are especially good at

incorporating site dependent costs, (T/F)______(2),

while the site selecting algorithms are primarily strong at trading off fixed versus variable costs,

(T/F)______(3).

In the Add heuristic for the discrete warehouse location problem, the warehouse or depot with the most

negative site dependent cost is evaluated next for possible establishment (addition), (T/F)______(4).

It is the recommended practice that the first supply chain design project in a corporation should be for

the strategic design of the supply chain, (T/F) _____(5).


The Geoffrion and Graves model for strategic logistics systems design enforces customer single

sourcing service constraints, (T/F)______(6),

and is an arc-based formation, (T/F)______(7).

The Kuehn and Hamburger model for strategic logistics systems design enforces customer single

sourcing service constraints, (T/F)______(8).

ColorJet Company

The ColorJet Company manufactures color inkjet printers. The manufacturing plant is located on the

West Coast of the United States and the company operates a regional distribution center on the East

Coast. You are responsible for selecting the least cost transportation mode between the manufacturing

plant and the distribution center. The possible transportation modes are rail, piggyback (truck on rail),

truck, and air. Various characteristics for each mode are given in the next table.

Table 9.9. Transportation Mode Selection Data

Characteristic Rail Piggyback Truck AirUnit Transportation Cost ($) 1 1.5 2 14Channel Transit Time (days) 21 14 5 1Transportation Batch Size 1500 360 360 50

The unit manufacturing cost of the printers is $300. The aggregate holding cost for all inventories for

one year is 30 % of the product value, i.e., 30 cents per dollar per year. The total customer demand

served from the distribution center is 18,000 printers per year. It is assumed that both production and

demand processes have a constant rate throughout the year. A year is equivalent to 365 days. The daily

demand for printers at the distribution center has a coefficient of variation equal to two. The safety

inventory for each mode in the distribution center is sufficient so that the probability of stock-out during

the lead-time is less than 5 % for that transportation mode. The annualized fixed warehouse cost is

equal to $250 per storage location and each storage location can hold 10 printers.

Determine the best transportation mode based on the total cost for this production-distribution system.

Show your results in a clear table (alternatives versus costs) and compute all costs on an annual basis.

Be sure to indicate the units for all numerical results.


Million Bubbles Company

The Million Bubbles Company manufactures Jacuzzi baths for customers in the continental United

States and Alaska. The manufacturing plant is located in Macon, Georgia, on the East Coast of the

United States and the company operates a regional distribution center in Seattle, Washington on the

West Coast of the United States. You are responsible for selecting the least cost transportation mode

between the manufacturing plant and the distribution center. The possible transportation modes are rail

or truck. Various characteristics for each mode are given in the next table.

Table 9.10. Transportation Mode Selection Data

Characteristic Rail TruckUnit Transportation Cost ($) 18 160Channel Transit Time (days) 15 5Transportation Batch Size 20 6

The unit manufacturing cost of the Jacuzzi baths is $1,500. Due to the annual style and color changes in

the Jacuzzi models, the aggregate holding cost for all inventories for one year is 90 % of the product

value, i.e., 90 cents per dollar per year. The total customer demand served from the distribution center is

360 Jacuzzi baths per year. It is assumed that both production and demand processes have a constant

average rate throughout the year. A year is equivalent to 360 days. The daily demand for baths at the

distribution center has a normal distribution with a mean of one bath and a standard deviation of 0.33

baths. The safety inventory for each mode in the distribution center is assumed to ensure a service level

equivalent to a probability of 99.5 % delivery out of inventory for the daily demand of Jacuzzi baths.

The annualized fixed warehouse cost is equal to $125 per bath.

Determine the best transportation mode based on the total cost for this production-distribution system.

Show your results in a clear table (alternatives versus costs) and compute all costs on an annual basis.

Be sure to indicate the units for all numerical results.

Smile Extension Assembly Operations in Warehouses

Consider the SMILE model by Wei and Goetschalckx for the design of strategic production-distribution

systems. Assume that some assembly operations can be performed in the last warehouse before the

products reach the customers. There are two intermediate products, denoted by A and B, and there are

two final products, denoted by C and D. The assembly equations for the final assembly are:


2 32 5 2

A B CB A

+ =+ =

D

In other words, the first assembly equation states that two units of product A plus three units of product

B are required to make one unit of product C.

Assembling one product C takes 9 minutes, assembling two units of product D takes 24 minutes. The

handling of one unit of product C from receiving through shipping takes 3 minutes. The handling of one

unit of product D from receiving through shipping takes 7 minutes. The handling of an assembled unit

of product C takes 1 minute. The handling of an assembled unit of product D takes 2 minutes. All

assembly and handling has to be done by a team of 10 equivalent workers. Each works 480 minutes per

shift. The cost per minute of labor spent on either handling or assembling products C and D is equal to

$1. Assembling product D requires the presence of a particular machine. This presence is denoted by

variable , equal to 1 if the machine is present, zero otherwise. The fixed cost per shift for

Machine N is equal to $100.

MachineN

1. Clearly define all your variables and parameters consistent with the

Wei and Goetschalckx model and list the legend.

2. Write down explicitly the flow conservation equations for the last

warehouse before the products reach the customers (i.e., without

summation signs and with numerical values for all parameters).

3. Write down explicitly the capacity constraint for the total throughput

of products C and D through this final warehouse (i.e., without

summation signs and with numerical values for all parameters).

4. Write the linkage or consistency constraints for assembly of product D

and Machine N

5. Write the cost function for the handling and assembly of products C

and D.


Recycling Operations

Consider the integrated model for the design of strategic logistics systems. SMILE is an example of

such a model. The following questions are related to extending this strategic logistics model to

incorporate for various additional constraints, commonly encountered in practice. In your answers,

clearly indicate the bounds of any summations signs you might have used and the number of constraints

of that particular type.

Using the notation developed in class for the SMILE model, write down the most compact constraint

which assures that at most one transportation channel between an origin facility i and destination facility

j is used, assuming there are many alternative or parallel transportation channels available between the

origin and destination facility. These alternatives are indexed by m. Give a clear definition of the

variables and parameters used in this constraint. Will there be many or few constraints of this type in

the extended model?

Assume that if a warehouse is used, then it must have a minimum throughput or material flow through

this warehouse. This minimum throughput is denoted by MinFlowjl for a warehouse of type l at site j.

Write down the most compact constraint which assures that if the warehouse of that type at that site is

used it will have at least the required minimum throughput. Give a clear definition of the variables and

parameters used in this constraint. Will there be many or few constraints of this type in the extended

model?

Consider the case where a raw material vendor i ships raw material p to a manufacturing plant j. The

production process at the manufacturing plant creates a finished product q and scrap material t. The

finished product is shipped to distribution center k. The scrap material is returned to the vendor from

which it came to be recycled at the raw material vendor. All shipments occur at a weekly frequency.

The amount of all material flows is expressed in tons per week. The production process at the plant has

a 20 % scrap rate, i.e., 20 percent of the incoming raw material is scrapped. All material flows are

carried by identical, company owned trucks which carry exclusively company materials. The materials

are heavy so all trucks are constrained by their weight capacity only. The company can use or deploy at

most V trucks. Write down all the required constraints to ensure that the SMILE model will find a

feasible solution. Use the notation developed in class and extend in a logical fashion when required.

Write the most compact constraints. Give a clear definition of the variables and parameters used in this


constraint. The formulation for this case depends strongly on the detailed assumptions that are made

regarding transportation. List clearly all your transportation assumptions.

Power Buy

The Savvy Customers (SC) chain of warehouse stores has been offered the opportunity for a special deal

by a national manufacturer of laundry detergent. If SC purchases the expected sales quantity for the

following quarter in one order and takes delivery of the complete quantity of the detergent at the

beginning of the quarter, the manufacturer is offering a 25 % discount on the purchase price. This

practice is commonly known as a “power buy”. For the traditional, staggered purchases and deliveries,

the only practical transportation mode between the manufacturing plant and the distribution center of SC

is by truck. When the complete quantity is transported and delivered as a single quantity, transportation

by rail has become an additional possible transportation mode. As the junior industrial engineer

employed by SC you are asked to evaluate the three alternatives and make a recommendation based on

which alternative has the lowest total cost.

The various quantities of detergent are given in pallets. The quarterly demand is 3000 pallets. The

purchasing price when ordering in multiple orders is $120 per pallet. The holding cost rate is 25 cents

on the dollar per quarter. The cost per pallet location in the warehouse is $16 per quarter. The

warehouse stores have an explicit “quantities are limited” customer service policy and the stores nor the

distribution center keep any safety inventory. The traditional periodic purchases occur once a week.

You can assume that there are 12 weeks and that there are 90 days in a quarter. The transportation cost

and transportation times for each of the three alternatives are given in the next table.

Powerbuy Rail Powerbuy Truck Multibuy Truck UnitsUnit Transportation Cost ($) 4 20 24 $/palletChannel Transit Time (days) 14 5 5 daysUnit Purchasing Cost $120.00 $/pallet

Summarize your calculations in a table using the traditional cost categories. Clearly separate and sum

different aggregate cost categories such as invariant, variable, and fixed costs. The costs in each

category must sum to the aggregate cost of the category. The sum of all aggregate costs must be equal

to the total cost. You can use rows of the table to perform intermediate calculations if this clarifies your

cost calculations. In the first column give the title of the cost, intermediate variable, or cost category, in

the second column give the formula you used to compute this cost, in the next three columns give the

computed cost for each of the three alternatives, and in the last column give the units for the variable or


cost in that row. You must specify the units to get credit for the numerical answers in that row. Execute

and display all cost calculations in whole dollars.

References 1. Benders, P., (1962), "Partitioning Procedures for Solving Mixed-Variables Programming

Problems," Numerische Mathematik, Vol. 4, pp. 238-252.

2. Canel, C. and Khumawala, B. M. (1997), “Multi-period international facilities location: An algorithm and application,” International Journal of Production Research, Vol. 35, No. 7, pp. 1891-1910.

3. Cohen, M. A. and A. Huchzermeier, (1999), “Global supply chain management: a survey of research and applications,” in Quantitative Models for Supply Chain Management, Tayur S. et al. (Eds.), Kluwer Academic Publishers, Boston, Massachusetts, pp 669-702.

4. Cohen, M. A. and H. L. Lee, (1985), “Manufacturing strategy: concepts and methods”, Chapter 5 in The Management of Productivity and Technology in Manufacturing, P. R. Kleindorfer (Editor), Plenum Press, NY, 153-188.

5. Cohen, M. A. and H. L. Lee, (1988), “Strategic analysis of integrated production-distribution systems: models and methods”, Operations Research, Vol. 36, No. 2, pp. 216-228.

6. Cohen, M. A. and H. L. Lee, (1989), “Resource deployment analysis of global manufacturing and distribution networks”, Journal of Manufacturing Operations Management, Vol. 2, pp. 81-104.

7. Cohen, M. A. and S. Moon, (1991), “An integrated plant loading model with economies of scale and scope,” European Journal of Operational Research, Vol. 50, No. 3, pp. 266-279.

8. Cohen, M. A., and P. R. Kleindorfer, (1993), "Creating Value through Operations: The Legacy of Elwood S. Buffa", in Perspectives in Operations Management (Essays in Honor of Elwood S. Buffa), R. K. Sarin, (Ed.), Kluwer Academic Publishers, Boston, Massachusetts, pp. 3-21.

9. Cohen, M. A., M. Fisher, and R. Jaikumar, (1989), “International manufacturing and distribution networks: a normative model framework”, in Managing International Manufacturing, K. Ferdows (Editor), North-Holland, Amsterdam, pp. 67-93.

10. Erlenkotter D., (1978). "A Dual-Based Procedure for Uncapacitated Facility Location". Operations Research, Vol. 26, No. 6, pp. 992-1009.

11. Fisher M. L., (1985). "An Applications Oriented Guide to Lagrangian Relaxation". Interfaces, Vol. 15, No. 2, pp. 10-21.

12. Francis, R. L., L. F. McGinnis, and J. A. White, 2nd Edition (1992). Facility Layout and Location: An Analytical Approach. Prentice-Hall, Englewood Cliffs, New Jersey.


13. Geoffrion A. M. and G. W. Graves, (1974). "Multicommodity distribution system design by Benders decomposition." Management Science, Vol. 20, No. 5, pp. 822-844.

14. Geoffrion A. M. and McBride, (1978). “Lagreangean Relaxation Applied to Capacitated Facility Location Problems.” Operations Research, Vol. 10, No. 1, pp. 40-47.

15. Geoffrion, A. M. and R. F. Powers, (1995), “20 Years of Strategic Distribution System Design: an Evolutionary Perspective," Interfaces, Vol. 25, No. 5, pp. 105-127.

16. Geoffrion, A. M., and R. F. Powers, (1980), “Facility location analysis is just the beginning (if you do it right)”, Interfaces 10/2, 22-30.

17. Geoffrion, A. M., G. W. Graves, and S. J. Lee, (1982). “A Management Support System for Distribution Planning.” INFOR 20, No. 4, pp. 287-314.

18. Geoffrion, A. M., G. W. Graves, and S. J. Lee, “Strategic Distribution System Planning: A Status Report,” in Studies in Operations Management, A. C. Hax, ed., North-Holland, Amsterdam, pp. 179-204.

19. Geoffrion, A. M., J. G. Morris, and S. T. Webster, (1995). “Distribution System Design,” in Facility Location: A Survey of Applications and Methods, Zvi Drezner (Editor), 1995, Springer Verlag, New York, New York.

20. Goetschalckx, M., (2000), “Strategic Network Planning,” in Supply Chain Management and Advanced Planning, Stadtler H. and C. Kilger (Eds.), Springer, Heidelberg, Germany.

21. Goetschalckx, M., G. Nemhauser, M. H. Cole, R. Wei, K. Dogan, and X. Zang, (1994), “Computer Aided Design of Industrial Logistic Systems,” in Proceedings of the Third Triennial Symposium on Transportation Analysis (TRISTAN III), Capri, Italy, pp. 151-178.

22. Kuehn, A.A., and Hamburger, M.J., "A heuristic program for locating warehouses", Management Science, 9, 643-666 (1963)

23. Lasdon, L. S., (1970). Optimization Theory for Large Systems. McMillan Publishing Co., New York, New York.

24. Lee, C., (1991), “An optimal algorithm for the multiproduct capacitated facility location problem with a choice of facility type,” Computers and Operational Research, Vol. 18, No. 2, pp. 167-182.

25. Lee, C., (1993), “A cross decomposition algorithm for a multiproduct-multitype facility location problem,” Computers and Operational Research, Vol. 20, No. 5, pp. 527-540.

26. Love R. F., J. G. Morris, and G. O. Wesolowsky, (1988). Facilities Location. Elsevier Science Publishing Co., New York, New York.

27. Mirchandani, P. B. and R. L. Francis, (1990). Discrete Location Theory. John Wiley & Sons, New York, New York.


28. Nemhauser G. L. and L. A. Wolsey, (1988). Integer and Combinatorial Optimization. John Wiley and Sons, Inc., New York, New York.

29. Park and G. P. Sharp, (1990). Engineering Economics.

30. Schmidt, G. and W. E. Wilhelm, (2000), “Strategic, tactical, and operational decisions in multi-national logistics networks: a review and discussion of modeling issues,” International Journal of Production Research, Vol. 38, No. 7, pp. 1501-1523.

31. Schrage, L., (1986). Linear, Integer, and Quadratic Programming with LINDO. The Scientific Press.

32. Stadtler, H. and C. Kilger, (2000), Supply Chain Management and Advanced Planning, Springer, Heidelberg, Germany.

33. Tayur, S., Ganeshan, R., and Magazine, M., (Eds.), (1999), Quantitative Models for Supply Chain Management, Kluwer Academic Publishers, Boston, Massachusetts.

34. Thomas, D. and Griffin, P. M., (1996), “Coordinated supply chain management," European Journal of Operational Research, Vol. 94, pp. 1-15.

35. Van Roy, T. J. and D. Erlenkotter, (1982), "A dual-based procedure for dynamic facility location," Management Science, Vol. 28, pp. 1091-1105.

36. Van Roy, T., (1983), “Cross decomposition for mixed integer programming,” Mathematical Programming, Vol. 25, pp. 46-63.

37. Van Roy, T., (1986). “A Cross Decomposition Algorithm for Capacitated Facility Location.” Operations Research, Vol. 34, No. 1, pp. 145-163.

38. Verter, V. and A. Dasci, (2001), “The plant location and flexible technology acquisition problem,” European Journal of Operational Research, (to appear).

39. Vidal C. and Goetschalckx, M. (1997), “Strategic Production-Distribution Models: A Critical Review with Emphasis on Global Supply Chain Models”, European Journal of Operational Research, Vol. 98, pp. 1-18.

40. Vidal C. and M. Goetschalckx, (2001), "A Global Supply Chain Model with Transfer Pricing and Transportation Cost Allocation," European Journal of Operational Research, Vol. 129, No. 1, pp. 134-158.

41. Vidal, C. and M. Goetschalckx, (1996), “The Role and Limitations of Quantitative Techniques in the Strategic Design of Global Logistics Systems”, CIBER Research Report 96-023, Georgia Institute of Technology. Accepted for publication in the special issue on Manufacturing in a Global Economy of the Journal of Technology and Forecasting and Social Change.

42. Vidal, C. and M. Goetschalckx, (2000), "Modeling the Impact of Uncertainties on Global Logistics Systems," Journal of Business Logistics, Vol. 21. No. 1, pp. 95-120.


chapter 9. supply chain models - isye | georgia … 9. supply chain models this is an introduction...

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